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Functional equation in reals

Source: 2008 Bulgarian Autumn Math Competition, Problem 12.3

March 18, 2022

functionalgebraFunctional equation in Rfunctional equation

Problem Statement

Find all continuous functions

f:\mathbb{R}\rightarrow \mathbb{R}

such that (f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y) \forall x,y\in \mathbb{R}